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Un - \textbf{campo vettoriale (tangente)} è una mappa liscia + \textbf{campo vettoriale tangente} è una mappa liscia $v : M \to \RR^k$ tale per cui: \[ \boxed{v(x) \in T_x M, \quad \forall x \in M.} @@ -2118,7 +2118,7 @@ \begin{definition}[Pettinabilità] Una varietà liscia, con o senza bordo, si dice \textbf{pettinabile} - se ammette un campo vettoriale mai nullo. + se ammette un campo vettoriale tangente mai nullo. \end{definition} \begin{theorem}[di pettinabilità della sfera] \label{thm:pettinabilità_sfera} @@ -2126,12 +2126,12 @@ \end{theorem} \begin{proof} - Se $n$ è dispari, un campo vettoriale mai nullo è il seguente: + Se $n$ è dispari, un campo vettoriale tangente mai nullo è il seguente: \[ f(x_1, x_2, \ldots, x_n, x_{n+1}) = (-x_2, x_1, \ldots, -x_{n+1}, x_n) \in x^\perp. \] Sia ora $n$ pari. - Supponiamo $v$ sia un campo vettoriale mai nullo. Senza + Supponiamo $v$ sia un campo vettoriale tangente mai nullo. Senza perdita di generalità, possiamo considerarlo unitario. Allora possiamo costruire un omotopia liscia dalla mappa antipodale $A$ a $\id_{S^n}$ nel seguente modo: @@ -2143,4 +2143,57 @@ ha grado $1$, mentre la mappa antipodale ha grado $(-1)^{n+1} = -1$ per il Lemma \ref{lem:grado_antipodale}. Quindi $v$ non può esistere per $n$ pari. \end{proof} + + \section{Indici di campi vettoriali} + + \subsection{Zero isolato e indice di un campo in uno zero} + + \begin{definition}[Zero isolato] + Sia $f : U \subseteq \RR^m \to \RR^m$ liscia con $U$ aperto. Allora + $z$ si dice \textbf{zero isolato} di $f$ se esiste un raggio $\eps > 0$ + tale per cui $f$ in $B_\eps(z)$ ammette come unico zero $z$. + \end{definition} + + \begin{remark}[L'indice è ben definito] + Sia $\eps > 0$ tale per cui $z$ è unico zero per $f$ in $B_\eps(z)$. + Sia $v_\eps : S^m \to \partial B_\eps(z)$ tale per cui: + \[ + v_\eps(x) = z + \eps x. + \] + Osserviamo che $v_\eps$ preserva l'orientazione. Consideriamo + \[ + \overline{f_\eps} \defeq \bigrestr{\frac{f}{\norm{f}}}{\partial B_\eps(z)}. + \] + Poiché $v_\eps$ preserva l'orientazione, $\deg(\overline{f_\eps}) = \deg(\overline{f_\eps} \circ v_\eps)$. + Scelto un altro $\eps'$, possiamo definire un'omotopia $H$ nel seguente modo: + \[ + H_t = \overline{f_{(1-t)\eps + \eps'}} \circ v_{(1-t)\eps + \eps'}. + \] + Allora, per il Teorema \ref{thm:fondamentale_grado_intero}: + \[ + \deg(\overline{f_\eps}) = \deg(\overline{f_\eps} \circ v_\eps) = \deg(\overline{f_{\eps'}} \circ v_{\eps'}) = \deg(\overline{f_{\eps'}}). + \] + \end{remark} + + \begin{definition}[Indice di $f$ in $z$] + Sia $f : U \subseteq \RR^m \to \RR^m$ liscia con $U$ aperto. Sia + $z$ uno zero isolato di $f$. Si definisce allora l'\textbf{indice di $f$ in $z$} + come: + \[ + \boxed{\ind(f, z) \defeq \deg(\overline{f_\eps}), \quad \overline{f_\eps} \defeq \bigrestr{\frac{f}{\norm{f}}}{\partial B_\eps(z)},} + \] + dove $\eps$ è un raggio tale per cui $z$ è unico zero in $B_\eps(z)$. + \end{definition} + + \begin{corollary}[Indice di $z^k$ in $0$] + Sia $v_k : \CC \cong \RR^2 \to \CC$ tale per cui + $v_k(z) = z^k$. Allora $\ind(v_k, 0) = k$. + \end{corollary} + + \begin{proof} + Segue immediatamente dal Lemma \ref{lem:grado_zk}. + \end{proof} + + \subsection{Lemma di Hopf} + \end{multicols*}